It is interesting what people are trying to guess the meaning of "Jack, I swear--" from their viewing of the movie; but, they never checked the words in Annie Proulx's short story. The part about buying the postcard is not in the movie either.
A few weeks later on the Saturday he threw all Stoutamire's dirty horse blankets into the back of his pickup and took them down to the Quik Stop Car Wash to turn the high-pressure spray on them. When the wet clean blankets were stowed in the truck bed he stepped into Higgins's gift shop and busied himself with the postcard rack.
"Ennis, what are you lookin for rootin through them postcards?" said Linda Higgins, throwing a sopping brown coffee filter into the garbage can. "Scene a Brokeback Mountain."
"Over in Fremont County?"
"No, north a here."
"I didn't order none a them. Let me get the order list. They got it I can get you a hunderd. I got a order some more cards anyway."
"One's enough," said Ennis. When it came -- thirty cents -- he pinned it up in his trailer, brass-headed tack in each corner. Below it he drove a nail and on the nail he hung the wire hanger and the two old shirts suspended from it. He stepped back and looked at the ensemble through a few stinging tears.
"Jack, I swear -- " he said, though Jack had never asked him to swear anything and was himself not the swearing kind.
What Ennis did not finish aloud was thought in words. Ennis swore an oath to, somewhat like one does in a courtroom or when taking public office, promising that he would never stop loving him.
In the cowboy, country, Native American and Jesus' way, when a man made an oath promise, he kept it no matter what. And he could swear something without the other person having to do anything.
Some people have confused cursing or using strong "adult" language, aka cussin', with swearing. They have even confused the commandment "Do not take the LORD's name in vain," with saying "Jesus" as a cuss word. But, "Jesus" was the name of a lot of people in the 1st Century AD. Actually, the name in Hebrew was "Joshua."
But, taking the LORD's name in vain had to do with making a promise and using the name of YHWH (the real name of the God of the Israelites/Jews/Hebrews as one's witness and then not keeping the promise.
Jesus amended that commandment and said "Don't swear by anything, in heaven or on the earth; let your 'yes' be 'yes' and your 'no' be 'no'."
We have politicans who put a hand on the Bible at the swearing-in ceremonies when taking the oath of office. And, what are they swearing on and by? God in heaven and a Bible on earth. When a politician does not do his best to keep his promises to his constituents when he actually has the wherewithall to do so, he has taken God's name in vain.